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Journal Cover Physica D: Nonlinear Phenomena
  [SJR: 1.048]   [H-I: 89]   [3 followers]  Follow
    
   Hybrid Journal Hybrid journal (It can contain Open Access articles)
   ISSN (Print) 0167-2789
   Published by Elsevier Homepage  [2817 journals]
  • Random walk centrality in interconnected multilayer networks
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Albert Solé-Ribalta, Manlio De Domenico, Sergio Gómez, Alex Arenas
      Real-world complex systems exhibit multiple levels of relationships. In many cases they require to be modeled as interconnected multilayer networks, characterizing interactions of several types simultaneously. It is of crucial importance in many fields, from economics to biology and from urban planning to social sciences, to identify the most (or the less) influent nodes in a network using centrality measures. However, defining the centrality of actors in interconnected complex networks is not trivial. In this paper, we rely on the tensorial formalism recently proposed to characterize and investigate this kind of complex topologies, and extend two well known random walk centrality measures, the random walk betweenness and closeness centrality, to interconnected multilayer networks. For each of the measures we provide analytical expressions that completely agree with numerically results.


      PubDate: 2016-04-29T18:42:49Z
       
  • On the evolution of scattering data under perturbations of the Toda
           lattice
    • Abstract: Publication date: Available online 25 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): D. Bilman, I. Nenciu
      We present the results of an analytical and numerical study of the long-time behavior for certain Fermi-Pasta-Ulam (FPU) lattices viewed as perturbations of the completely integrable Toda lattice. Our main tools are the direct and inverse scattering transforms for doubly-infinite Jacobi matrices, which are well-known to linearize the Toda flow. We focus in particular on the evolution of the associated scattering data under the perturbed vs. the unperturbed equations. We find that the eigenvalues present initially in the scattering data converge to new, slightly perturbed eigenvalues under the perturbed dynamics of the lattice equation. To these eigenvalues correspond solitary waves that emerge from the solitons in the initial data. We also find that new eigenvalues emerge from the continuous spectrum as the lattice system is let to evolve under the perturbed dynamics.


      PubDate: 2016-04-29T18:42:49Z
       
  • Inverse scattering transform for the defocusing nonlinear Schrödinger
           equation with fully asymmetric non-zero boundary conditions
    • Abstract: Publication date: Available online 26 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Gino Biondini, Emily Fagerstrom, Barbara Prinari
      We formulate the inverse scattering transform (IST) for the defocusing nonlinear Schrödinger (NLS) equation with fully asymmetric non-zero boundary conditions (i.e., when the limiting values of the solution at space infinities have different non-zero modulii). The theory is formulated without making use of Riemann surfaces, and instead by dealing explicitly with the branched nature of the eigenvalues of the associated scattering problem. For the direct problem, we give explicit single-valued definitions of the Jost eigenfunctions and scattering coefficients over the whole complex plane, and we characterize their discontinuous behavior across the branch cut arising from the square root behavior of the corresponding eigenvalues. We pose the inverse problem as a Riemann Hilbert Problem on an open contour, and we reduce the problem to a standard set of linear integral equations. Finally, for comparison purposes, we present the single-sheet, branch cut formulation of the inverse scattering transform for the initial value problem with symmetric (equimodular) non-zero boundary conditions, as well as for the initial value problem with one-sided non-zero boundary conditions, and we also briefly describe the formulation of the inverse scattering transform when a different choice is made for the location of the branch cuts.


      PubDate: 2016-04-29T18:42:49Z
       
  • Absolute stability and synchronization in neural field models with
           transmission delays
    • Abstract: Publication date: Available online 26 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Chiu-Yen Kao, Chih-Wen Shih, Chang-Hong Wu
      Neural fields model macroscopic parts of the cortex which involve several populations of neurons. We consider a class of neural field models which are represented by integro-differential equations with transmission time delays which are space-dependent. The considered domains underlying the systems can be bounded or unbounded. A new approach, called sequential contracting, instead of the conventional Lyapunov functional technique, is employed to investigate the global dynamics of such systems. Sufficient conditions for the absolute stability and synchronization of the systems are established. Several numerical examples are presented to demonstrate the theoretical results.


      PubDate: 2016-04-29T18:42:49Z
       
  • Topological microstructure analysis using persistence landscapes
    • Abstract: Publication date: Available online 28 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Paweł Dłotko, Thomas Wanner
      Phase separation mechanisms can produce a variety of complicated and intricate microstructures, which often can be difficult to characterize in a quantitative way. In recent years, a number of novel topological metrics for microstructures have been proposed, which measure essential connectivity information and are based on techniques from algebraic topology. Such metrics are inherently computable using computational homology, provided the microstructures are discretized using a thresholding process. However, while in many cases the thresholding is straightforward, noise and measurement errors can lead to misleading metric values. In such situations, persistence landscapes have been proposed as a natural topology metric. Common to all of these approaches is the enormous data reduction, which passes from complicated patterns to discrete information. It is therefore natural to wonder what type of information is actually retained by the topology. In the present paper, we demonstrate that averaged persistence landscapes can be used to recover central system information in the Cahn–Hilliard theory of phase separation. More precisely, we show that topological information of evolving microstructures alone suffices to accurately detect both concentration information and the actual decomposition stage of a data snapshot. Considering that persistent homology only measures discrete connectivity information, regardless of the size of the topological features, these results indicate that the system parameters in a phase separation process affect the topology considerably more than anticipated. We believe that the methods discussed in this paper could provide a valuable tool for relating experimental data to model simulations.


      PubDate: 2016-04-29T18:42:49Z
       
  • Correlation functions of the KdV hierarchy and applications to
           intersection numbers over M¯g,n
    • Abstract: Publication date: Available online 29 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Marco Bertola, Boris Dubrovin, Di Yang
      We derive an explicit generating function of correlations functions of an arbitrary tau-function of the KdV hierarchy. In particular applications, our formulation gives closed formulæ of a new type for the generating series of intersection numbers of ψ -classes as well as of mixed ψ - and κ -classes in full genera.


      PubDate: 2016-04-29T18:42:49Z
       
  • Nonlinear Dynamics on Interconnected Networks
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Alex Arenas, Manlio De Domenico



      PubDate: 2016-04-29T18:42:49Z
       
  • On degree–degree correlations in multilayer networks
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Guilherme Ferraz de Arruda, Emanuele Cozzo, Yamir Moreno, Francisco A. Rodrigues
      We propose a generalization of the concept of assortativity based on the tensorial representation of multilayer networks, covering the definitions given in terms of Pearson and Spearman coefficients. Our approach can also be applied to weighted networks and provides information about correlations considering pairs of layers. By analyzing the multilayer representation of the airport transportation network, we show that contrasting results are obtained when the layers are analyzed independently or as an interconnected system. Finally, we study the impact of the level of assortativity and heterogeneity between layers on the spreading of diseases. Our results highlight the need of studying degree–degree correlations on multilayer systems, instead of on aggregated networks.


      PubDate: 2016-04-29T18:42:49Z
       
  • Systemic risk in multiplex networks with asymmetric coupling and threshold
           feedback
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Rebekka Burkholz, Matt V. Leduc, Antonios Garas, Frank Schweitzer
      We study cascades on a two-layer multiplex network, with asymmetric feedback that depends on the coupling strength between the layers. Based on an analytical branching process approximation, we calculate the systemic risk measured by the final fraction of failed nodes on a reference layer. The results are compared with the case of a single layer network that is an aggregated representation of the two layers. We find that systemic risk in the two-layer network is smaller than in the aggregated one only if the coupling strength between the two layers is small. Above a critical coupling strength, systemic risk is increased because of the mutual amplification of cascades in the two layers. We even observe sharp phase transitions in the cascade size that are less pronounced on the aggregated layer. Our insights can be applied to a scenario where firms decide whether they want to split their business into a less risky core business and a more risky subsidiary business. In most cases, this may lead to a drastic increase of systemic risk, which is underestimated in an aggregated approach.


      PubDate: 2016-04-29T18:42:49Z
       
  • Network bipartivity and the transportation efficiency of European
           passenger airlines
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Ernesto Estrada, Jesús Gómez-Gardeñes
      The analysis of the structural organization of the interaction network of a complex system is central to understand its functioning. Here, we focus on the analysis of the bipartivity of graphs. We first introduce a mathematical approach to quantify bipartivity and show its implementation in general and random graphs. Then, we tackle the analysis of the transportation networks of European airlines from the point of view of their bipartivity and observe significant differences between traditional and low cost carriers. Bipartivity shows also that alliances and major mergers of traditional airlines provide a way to reduce bipartivity which, in its turn, is closely related to an increase of the transportation efficiency.


      PubDate: 2016-04-29T18:42:49Z
       
  • Asymptotic periodicity in networks of degrade-and-fire oscillators
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Alex Blumenthal, Bastien Fernandez
      Networks of coupled degrade-and-fire (DF) oscillators are simple dynamical models of assemblies of interacting self-repressing genes. For mean-field interactions, which most mathematical studies have assumed so far, every trajectory must approach a periodic orbit. Moreover, asymptotic cluster distributions can be computed explicitly in terms of coupling intensity, and a massive collection of distributions collapses when this intensity passes a threshold. Here, we show that most of these dynamical features persist for an arbitrary coupling topology. In particular, we prove that, in any system of DF oscillators for which in and out coupling weights balance, trajectories with reasonable firing sequences must be asymptotically periodic, and periodic orbits are uniquely determined by their firing sequence. In addition to these structural results, illustrative examples are presented, for which the dynamics can be entirely described.


      PubDate: 2016-04-29T18:42:49Z
       
  • Cascades in interdependent flow networks
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Antonio Scala, Pier Giorgio De Sanctis Lucentini, Guido Caldarelli, Gregorio D’Agostino
      In this manuscript, we investigate the abrupt breakdown behavior of coupled distribution grids under load growth. This scenario mimics the ever-increasing customer demand and the foreseen introduction of energy hubs interconnecting the different energy vectors. We extend an analytical model of cascading behavior due to line overloads to the case of interdependent networks and find evidence of first order transitions due to the long-range nature of the flows. Our results indicate that the foreseen increase in the couplings between the grids has two competing effects: on the one hand, it increases the safety region where grids can operate without withstanding systemic failures; on the other hand, it increases the possibility of a joint systems’ failure.
      Graphical abstract image

      PubDate: 2016-04-29T18:42:49Z
       
  • Erosion of synchronization: Coupling heterogeneity and network structure
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Per Sebastian Skardal, Dane Taylor, Jie Sun, Alex Arenas
      We study the dynamics of network-coupled phase oscillators in the presence of coupling frustration. It was recently demonstrated that in heterogeneous network topologies, the presence of coupling frustration causes perfect phase synchronization to become unattainable even in the limit of infinite coupling strength. Here, we consider the important case of heterogeneous coupling functions and extend previous results by deriving analytical predictions for the total erosion of synchronization. Our analytical results are given in terms of basic quantities related to the network structure and coupling frustration. In addition to fully heterogeneous coupling, where each individual interaction is allowed to be distinct, we also consider partially heterogeneous coupling and homogeneous coupling in which the coupling functions are either unique to each oscillator or identical for all network interactions, respectively. We demonstrate the validity of our theory with numerical simulations of multiple network models, and highlight the interesting effects that various coupling choices and network models have on the total erosion of synchronization. Finally, we consider some special network structures with well-known spectral properties, which allows us to derive further analytical results.


      PubDate: 2016-04-29T18:42:49Z
       
  • Contact-based model for strategy updating and evolution of cooperation
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Jianlei Zhang, Zengqiang Chen
      To establish an available model for the astoundingly strategy decision process of players is not easy, sparking heated debate about the related strategy updating rules is intriguing. Models for evolutionary games have traditionally assumed that players imitate their successful partners by the comparison of respective payoffs, raising the question of what happens if the game information is not easily available. Focusing on this yet-unsolved case, the motivation behind the work presented here is to establish a novel model for the updating of states in a spatial population, by detouring the required payoffs in previous models and considering much more players’ contact patterns. It can be handy and understandable to employ switching probabilities for determining the microscopic dynamics of strategy evolution. Our results illuminate the conditions under which the steady coexistence of competing strategies is possible. These findings reveal that the evolutionary fate of the coexisting strategies can be calculated analytically, and provide novel hints for the resolution of cooperative dilemmas in a competitive context. We hope that our results have disclosed new explanations about the survival and coexistence of competing strategies in structured populations.


      PubDate: 2016-04-29T18:42:49Z
       
  • Consensus dynamics on random rectangular graphs
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Ernesto Estrada, Matthew Sheerin
      A random rectangular graph (RRG) is a generalization of the random geometric graph (RGG) in which the nodes are embedded into a rectangle with side lengths a and b = 1 / a , instead of on a unit square [ 0 , 1 ] 2 . Two nodes are then connected if and only if they are separated at a Euclidean distance smaller than or equal to a certain threshold radius r . When a = 1 the RRG is identical to the RGG. Here we apply the consensus dynamics model to the RRG. Our main result is a lower bound for the time of consensus, i.e., the time at which the network reaches a global consensus state. To prove this result we need first to find an upper bound for the algebraic connectivity of the RRG, i.e., the second smallest eigenvalue of the combinatorial Laplacian of the graph. This bound is based on a tight lower bound found for the graph diameter. Our results prove that as the rectangle in which the nodes are embedded becomes more elongated, the RRG becomes a ’large-world’, i.e., the diameter grows to infinity, and a poorly-connected graph, i.e., the algebraic connectivity decays to zero. The main consequence of these findings is the proof that the time of consensus in RRGs grows to infinity as the rectangle becomes more elongated. In closing, consensus dynamics in RRGs strongly depend on the geometric characteristics of the embedding space, and reaching the consensus state becomes more difficult as the rectangle is more elongated.


      PubDate: 2016-04-29T18:42:49Z
       
  • Interplay between consensus and coherence in a model of interacting
           opinions
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Federico Battiston, Andrea Cairoli, Vincenzo Nicosia, Adrian Baule, Vito Latora
      The formation of agents’ opinions in a social system is the result of an intricate equilibrium among several driving forces. On the one hand, the social pressure exerted by peers favors the emergence of local consensus. On the other hand, the concurrent participation of agents to discussions on different topics induces each agent to develop a coherent set of opinions across all the topics in which he/she is active. Moreover, the pervasive action of external stimuli, such as mass media, pulls the entire population towards a specific configuration of opinions on different topics. Here we propose a model in which agents with interrelated opinions, interacting on several layers representing different topics, tend to spread their own ideas to their neighborhood, strive to maintain internal coherence, due to the fact that each agent identifies meaningful relationships among its opinions on the different topics, and are at the same time subject to external fields, resembling the pressure of mass media. We show that the presence of heterogeneity in the internal coupling assigned by agents to their different opinions allows to obtain states with mixed levels of consensus, still ensuring that all the agents attain a coherent set of opinions. Furthermore, we show that all the observed features of the model are preserved in the presence of thermal noise up to a critical temperature, after which global consensus is no longer attainable. This suggests the relevance of our results for real social systems, where noise is inevitably present in the form of information uncertainty and misunderstandings. The model also demonstrates how mass media can be effectively used to favor the propagation of a chosen set of opinions, thus polarizing the consensus of an entire population.


      PubDate: 2016-04-29T18:42:49Z
       
  • Quasi-steady state reduction for compartmental systems
    • Abstract: Publication date: Available online 21 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Alexandra Goeke, Christian Lax
      We present a method to determine an asymptotic reduction (in the sense of Tikhonov and Fenichel) for singularly perturbed compartmental systems in the presence of slow transport. It turns out that the reduction can be derived from the individual interaction terms alone. We apply the result to spatially discretized reaction-diffusion systems and obtain (based on the reduced discretized systems) a heuristic to reduce reaction-diffusion systems in presence of slow diffusion.


      PubDate: 2016-04-24T18:22:14Z
       
  • Dispersive shock waves and modulation theory
    • Abstract: Publication date: Available online 20 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): G.A. El, M.A. Hoefer
      There is growing physical and mathematical interest in the hydrodynamics of dissipationless/dispersive media. Since G. B. Whitham’s seminal publication fifty years ago that ushered in the mathematical study of dispersive hydrodynamics, there has been a significant body of work in this area. However, there has been no comprehensive survey of the field of dispersive hydrodynamics. Utilizing Whitham’s averaging theory as the primary mathematical tool, we review the rich mathematical developments over the past fifty years with an emphasis on physical applications. The fundamental, large scale, coherent excitation in dispersive hydrodynamic systems is an expanding, oscillatory dispersive shock wave or DSW. Both the macroscopic and microscopic properties of DSWs are analyzed in detail within the context of the universal, integrable, and foundational models for uni-directional (Korteweg–de Vries equation) and bi-directional (Nonlinear Schrödinger equation) dispersive hydrodynamics. A DSW fitting procedure that does not rely upon integrable structure yet reveals important macroscopic DSW properties is described. DSW theory is then applied to a number of physical applications: superfluids, nonlinear optics, geophysics, and fluid dynamics. Finally, we survey some of the more recent developments including non-classical DSWs, DSW interactions, DSWs in perturbed and inhomogeneous environments, and two-dimensional, oblique DSWs.


      PubDate: 2016-04-24T18:22:14Z
       
  • The extended Estabrook-Wahlquist method
    • Abstract: Publication date: Available online 21 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): S. Roy Choudhury, Matthew Russo
      Variable Coefficient Korteweg de Vries (vcKdV), modified Korteweg de Vries (vcMKdV), and nonlinear Schröedinger (NLS) equations have a long history dating from their derivation in various applications. A technique based on extended Lax Pairs has been devised recently to derive time-and-space-dependent-coefficient generalizations of various such Lax-integrable NLPDE hierarchies, which are thus more general than almost all cases considered earlier via methods such as the Painlevé Test, Bell Polynomials, and similarity methods. However, this technique, although operationally effective, has the significant disadvantage that, for any integrable system with spatiotemporally varying coefficients, one must ‘guess’ a generalization of the structure of the known Lax Pair for the corresponding system with constant coefficients. Motivated by the somewhat arbitrary nature of the above procedure, we embark in this paper on an attempt to systematize the derivation of Lax-integrable systems with variable coefficients. We consider the Estabrook-Wahlquist (EW) prolongation technique, a relatively self-consistent procedure requiring little prior information. However, this immediately requires that the technique be significantly generalized in several ways, including solving matrix partial differential equations instead of algebraic ones as the structure of the Lax Pair is systematically computed, and also in solving the constraint equations to deduce the explicit forms for various ‘coefficient’ matrices. The new and extended EW technique which results is illustrated by algorithmically deriving generalized Lax-integrable versions of the NLS, generalized fifth-order KdV, MKdV, and derivative nonlinear Schröedinger (DNLS) equations. We also show how this method correctly excludes the existence of a nontrivial Lax pair for a nonintegrable NLPDE such as the variable-coefficient cubic-quintic NLS.


      PubDate: 2016-04-24T18:22:14Z
       
  • Cellular replication limits in the Luria-Delbrück mutation model
    • Abstract: Publication date: Available online 19 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Ignacio A. Rodriguez-Brenes, Dominik Wodarz, Natalia L. Komarova
      Originally developed to elucidate the mechanisms of natural selection in bacteria, the Luria-Delbrück model assumed that cells are intrinsically capable of dividing an unlimited number of times. This assumption however, is not true for human somatic cells which undergo replicative senescence. Replicative senescence is thought to act as a mechanism to protect against cancer and the escape from it is a rate-limiting step in cancer progression. Here we introduce a Luria-Delbrück model that explicitly takes into account cellular replication limits in the wild type cell population and models the emergence of mutants that escape replicative senescence. We present results on the mean, variance, distribution, and asymptotic behavior of the mutant population in terms of three classical formulations of the problem. More broadly the paper introduces the concept of incorporating replicative limits as part of the Luria-Delbrück mutational framework. Guidelines to extend the theory to include other types of mutations and possible applications to the modeling of telomere crisis and fluctuation analysis are also discussed.


      PubDate: 2016-04-19T18:05:31Z
       
  • The Poincaré-Bendixson Theorem and the non-linear Cauchy-Riemann
           equations
    • Abstract: Publication date: Available online 18 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): J.B. van den Berg, S. Munaò, R.C.A.M. Vandervorst
      In Fiedler and Mallet-Paret (1989) prove a version of the classical Poincaré-Bendixson Theorem for scalar parabolic equations. We prove that a similar result holds for bounded solutions of the non-linear Cauchy-Riemann equations. The latter is an application of an abstract theorem for flows with a(n) (unbounded) discrete Lyapunov function.


      PubDate: 2016-04-19T18:05:31Z
       
  • On the generation of dispersive shock waves
    • Abstract: Publication date: Available online 18 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Peter D. Miller
      We review various methods for the analysis of initial-value problems for integrable dispersive equations in the weak-dispersion or semiclassical regime. Some methods are sufficiently powerful to rigorously explain the generation of modulated wavetrains, so-called dispersive shock waves, as the result of shock formation in a limiting dispersionless system. They also provide a detailed description of the solution near caustic curves that delimit dispersive shock waves, revealing fascinating universal wave patterns.


      PubDate: 2016-04-19T18:05:31Z
       
  • A trajectory-free framework for analysing multiscale systems
    • Abstract: Publication date: Available online 19 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Gary Froyland, Georg A. Gottwald, Andy Hammerlindl
      We develop algorithms built around properties of the transfer operator and Koopman operator which 1) test for possible multiscale dynamics in a given dynamical system, 2) estimate the magnitude of the time-scale separation, and finally 3) distill the reduced slow dynamics on a suitably designed subspace. By avoiding trajectory integration, the developed techniques are highly computationally efficient. We corroborate our findings with numerical simulations of a test problem.


      PubDate: 2016-04-19T18:05:31Z
       
  • Multi-soliton, multi-breather and higher order rogue wave solutions to the
           complex short pulse equation
    • Abstract: Publication date: Available online 19 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Liming Ling, Bao-Feng Feng, Zuonong Zhu
      In the present paper, we are concerned with the general analytic solutions to the complex short pulse (CSP) equation including soliton, breather and rogue wave solutions. With the aid of a generalized Darboux transformation, we construct the N -bright soliton solution in a compact determinant form, the N -breather solution including the Akhmediev breather and a general higher order rogue wave solution. The first and second order rogue wave solutions are given explicitly and analyzed. The asymptotic analysis is performed rigourously for both the N -soliton and the N -breather solutions. All three forms of the analytical solutions admit either smoothed-, cusped- or looped-type ones for the CSP equation depending on the parameters. It is noted that, due to the reciprocal (hodograph) transformation, the rogue wave solution to the CSP equation can be a smoothed, cusponed or a looped one, which is different from the rogue wave solution found so far.


      PubDate: 2016-04-19T18:05:31Z
       
  • Dynamics of curved fronts in systems with power-law memory
    • Abstract: Publication date: Available online 13 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): M. Abu Hamed, A.A. Nepomnyashchy
      The dynamics of a curved front in a plane between two stable phases with equal potentials is modeled via two-dimensional fractional in time partial differential equation. A closed equation governing a slow motion of a small-curvature front is derived and applied for two typical examples of the potential function. Approximate axisymmetric and non-axisymmetric solutions are obtained.


      PubDate: 2016-04-14T17:49:57Z
       
  • Geometric phase in the Hopf bundle and the stability of non-linear waves
    • Abstract: Publication date: Available online 13 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Colin J. Grudzien, Thomas J. Bridges, Christopher K.R.T. Jones
      We develop a stability index for the travelling waves of non-linear reaction diffusion equations using the geometric phase induced on the Hopf bundle S 2 n − 1 ⊂ C n . This can be viewed as an alternative formulation of the winding number calculation of the Evans function, whose zeroes correspond to the eigenvalues of the linearization of reaction diffusion operators about the wave. The stability of a travelling wave can be determined by the existence of eigenvalues of positive real part for the linear operator. Our method of geometric phase for locating and counting eigenvalues is inspired by the numerical results in Way’s Dynamics in the Hopf bundle, the geometric phase and implications for dynamical systems Way (2009). We provide a detailed proof of the relationship between the phase and eigenvalues for dynamical systems defined on C 2 and sketch the proof of the method of geometric phase for C n and its generalization to boundary-value problems. Implementing the numerical method, modified from Way (2009), we conclude with open questions inspired from the results.


      PubDate: 2016-04-14T17:49:57Z
       
  • Nonlinear random optical waves: Integrable turbulence, rogue waves and
           intermittency
    • Abstract: Publication date: Available online 8 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Stéphane Randoux, Pierre Walczak, Miguel Onorato, Pierre Suret
      We examine the general question of statistical changes experienced by ensembles of nonlinear random waves propagating in systems ruled by integrable equations. In our study that enters within the framework of integrable turbulence, we specifically focus on optical fiber systems accurately described by the integrable one-dimensional nonlinear Schrödinger equation. We consider random complex fields having a gaussian statistics and an infinite extension at initial stage. We use numerical simulations with periodic boundary conditions and optical fiber experiments to investigate spectral and statistical changes experienced by nonlinear waves in focusing and in defocusing propagation regimes. As a result of nonlinear propagation, the power spectrum of the random wave broadens and takes exponential wings both in focusing and in defocusing regimes. Heavy-tailed deviations from gaussian statistics are observed in focusing regime while low-tailed deviations from gaussian statistics are observed in defocusing regime. After some transient evolution, the wave system is found to exhibit a statistically stationary state in which neither the probability density function of the wave field nor the spectrum change with the evolution variable. Separating fluctuations of small scale from fluctuations of large scale both in focusing and defocusing regime, we reveal the phenomenon of intermittency; i.e., small scales are characterized by large heavy-tailed deviations from Gaussian statistics, while the large ones are almost Gaussian.


      PubDate: 2016-04-09T17:40:32Z
       
  • Primitive potentials and bounded solutions of the KdV equation
    • Abstract: Publication date: Available online 8 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): S. Dyachenko, D. Zakharov, V. Zakharov
      We construct a broad class of bounded potentials of the one-dimensional Schrödinger operator that have the same spectral structure as periodic finite-gap potentials, but that are neither periodic nor quasi-periodic. Such potentials, which we call primitive, are non-uniquely parametrized by a pair of positive Hölder continuous functions defined on the allowed bands. Primitive potentials are constructed as solutions of a system of singular integral equations, which can be efficiently solved numerically. Simulations show that these potentials can have a disordered structure. Primitive potentials generate a broad class of bounded non-vanishing solutions of the KdV hierarchy, and we interpret them as an example of integrable turbulence in the framework of the KdV equation.


      PubDate: 2016-04-09T17:40:32Z
       
  • Nonlinear optical vibrations of single-walled carbon nanotubes. 1. Energy
           exchange and localization of low-frequency oscillations
    • Abstract: Publication date: Available online 4 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): V.V. Smirnov, L.I. Manevitch, M. Strozzi, F. Pellicano
      We present the results of analytical study and molecular dynamics simulation of low energy nonlinear non-stationary dynamics of single-walled carbon nanotubes (CNTs). New phenomena of intense energy exchange between different parts of CNT and weak energy localization in the excited part of CNT are analytically predicted in the framework of the continuum shell theory. Their origin is clarified by means of the concept of Limiting Phase Trajectory, and the analytical results are confirmed by the molecular dynamics simulation of simply supported CNTs.


      PubDate: 2016-04-04T17:26:25Z
       
  • Existence and stability of PT-symmetric states in nonlinear
           two-dimensional square lattices
    • Abstract: Publication date: Available online 2 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Haitao Xu, P.G. Kevrekidis, Dmitry E. Pelinovsky
      Solitons and vortices symmetric with respect to simultaneous parity ( P ) and time reversing ( T ) transformations are considered on the square lattice in the framework of the discrete nonlinear Schrödinger equation. The existence and stability of such PT -symmetric configurations is analyzed in the limit of weak coupling between the lattice sites, when predictions on the elementary cell of a square lattice (i.e., a single square) can be extended to a large (yet finite) array of lattice cells. In particular, we find all examined vortex configurations are unstable with respect to small perturbations while a branch extending soliton configurations is spectrally stable. Our analytical predictions are found to be in good agreement with numerical computations.


      PubDate: 2016-04-04T17:26:25Z
       
  • Inertial effects on thin-film wave structures with imposed surface shear
           on an inclined plane
    • Abstract: Publication date: Available online 21 March 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): M. Sivapuratharasu, S. Hibberd, M.E. Hubbard, H. Power
      This study provides an extended approach to the mathematical simulation of thin-film flow on a flat inclined plane relevant to flows subject to high surface shear. Motivated by modelling thin-film structures within an industrial context, wave structures are investigated for flows with moderate inertial effects and small film depth aspect ratio ε . Approximations are made assuming a Reynolds number, Re ∼ O ( ε − 1 ) and depth-averaging used to simplify the governing Navier-Stokes equations. A parallel Stokes flow is expected in the absence of any wave disturbance and a generalisation for the flow is based on a local quadratic profile. This approch provides a more general system which includes inertial effects and is solved numerically. Flow structures are compared with studies for Stokes flow in the limit of negligible inertial effects. Both two-tier and three-tier wave disturbances are used to study film profile evolution. A parametric study is provided for wave disturbances with increasing film Reynolds number. An evaluation of standing wave and transient film profiles is undertaken and identifies new profiles not previously predicted when inertial effects are neglected.


      PubDate: 2016-03-25T13:22:38Z
       
  • Modulational instability in nonlinear nonlocal equations of regularized
           long wave type
    • Abstract: Publication date: Available online 18 March 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Vera Mikyoung Hur, Ashish Kumar Pandey
      We study the stability and instability of periodic traveling waves in the vicinity of the origin in the spectral plane, for equations of Benjamin-Bona-Mahony (BBM) and regularized Boussinesq types permitting nonlocal dispersion. We extend recent results for equations of Korteweg-de Vries type and derive modulational instability indices as functions of the wave number of the underlying wave. We show that a sufficiently small, periodic traveling wave of the BBM equation is spectrally unstable to long wavelength perturbations if the wave number is greater than a critical value and a sufficiently small, periodic traveling wave of the regularized Boussinesq equation is stable to square integrable perturbations.


      PubDate: 2016-03-21T13:19:25Z
       
  • Jump bifurcations in some degenerate planar piecewise linear differential
           systems with three zones
    • Abstract: Publication date: Available online 11 March 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Rodrigo Euzébio, Rubens Pazim, Enrique Ponce
      We consider continuous piecewise-linear differential systems with three zones where the central one is degenerate, that is, the determinant of its linear part vanishes. By moving one parameter which is associated to the equilibrium position, we detect some new bifurcations exhibiting jump transitions both in the equilibrium location and in the appearance of limit cycles. In particular, we introduce the scabbard bifurcation, characterized by the birth of a limit cycle from a continuum of equilibrium points. Some of the studied bifurcations are detected, after an appropriate choice of parameters, in a piecewise linear Morris-Lecar model for the activity of a single neuron activity, which is usually considered as a reduction of the celebrated Hodgkin-Huxley equations.


      PubDate: 2016-03-16T13:03:26Z
       
  • A numerical method for computing initial conditions of Lagrangian
           invariant tori using the frequency map
    • Abstract: Publication date: Available online 10 March 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Alejandro Luque, Jordi Villanueva
      We present a numerical method for computing initial conditions of Lagrangian quasi-periodic invariant tori of Hamiltonian systems and symplectic maps. Such initial conditions are found by solving, using the Newton method, a nonlinear system obtained by imposing suitable conditions on the frequency map. The basic tool is a newly developed methodology to perform the frequency analysis of a discrete quasi-periodic signal, allowing to compute frequencies and their derivatives with respect to parameters. Roughly speaking, this method consists in computing suitable weighted averages of the iterates of the signal and using the Richardson extrapolation method. The proposed approach performs with high accuracy at a moderate computational cost. We illustrate the method by considering a discrete FPU model and the vicinity of the point L 4 in a RTBP.


      PubDate: 2016-03-11T12:45:40Z
       
  • Effective integration of ultra-elliptic solutions of the focusing
           nonlinear Schrödinger equation
    • Abstract: Publication date: Available online 9 March 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): O.C. Wright
      An effective integration method based on the classical solution of the Jacobi inversion problem, using Kleinian ultra-elliptic functions and Riemann theta functions, is presented for the quasi-periodic two-phase solutions of the focusing cubic nonlinear Schrödinger equation. Each two-phase solution with real quasi-periods forms a two-real-dimensional torus, modulo a circle of complex-phase factors, expressed as a ratio of theta functions associated with the Riemann surface of the invariant spectral curve. The initial conditions of the Dirichlet eigenvalues satisfy reality conditions which are explicitly parametrized by two physically-meaningful real variables: the squared modulus and a scalar multiple of the wavenumber. Simple new formulas for the maximum modulus and the minimum modulus are obtained in terms of the imaginary parts of the branch points of the Riemann surface.


      PubDate: 2016-03-11T12:45:40Z
       
  • Nonlinear Schrödinger equations with a multiple-well potential and a
           Stark-type perturbation
    • Abstract: Publication date: Available online 8 March 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Andrea Sacchetti
      A Bose-Einstein condensate (BEC) confined in a one-dimensional lattice under the effect of an external homogeneous field is described by the Gross-Pitaevskii equation. Here we prove that such an equation can be reduced, in the semiclassical limit and in the case of a lattice with a finite number of wells, to a finite-dimensional discrete nonlinear Schrödinger equation. Then, by means of numerical experiments we show that the BEC’s center of mass exhibits an oscillating behavior with modulated amplitude; in particular, we show that the oscillating period actually depends on the shape of the initial wavefunction of the condensate as well as on the strength of the nonlinear term. This fact opens a question concerning the validity of a method proposed for the determination of the gravitational constant by means of the measurement of the oscillating period.


      PubDate: 2016-03-11T12:45:40Z
       
  • Dressing method for the vector sine-Gordon equation and its soliton
           interactions
    • Abstract: Publication date: Available online 9 March 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Alexander V. Mikhailov, Georgios Papamikos, Jing Ping Wang
      In this paper, we develop the dressing method to study the exact solutions for the vector sine-Gordon equation. The explicit formulas for one kink and one breather are derived. The method can be used to construct multi-soliton solutions. Two soliton interactions are also studied. The formulas for position shift of the kink and position and phase shifts of the breather are given. These quantities only depend on the pole positions of the dressing matrices.


      PubDate: 2016-03-11T12:45:40Z
       
  • Three–dimensional representations of the tube manifolds of the
           planar restricted three–body problem
    • Abstract: Publication date: Available online 10 March 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Elena Lega, Massimiliano Guzzo
      The stable and unstable manifolds of the Lyapunov orbits of the Lagrangian equilibrium points L1, L2 play a key role in the understanding of the complicated dynamics of the circular restricted three–body problem. By developing a recent technique of computation of the stable and unstable manifolds, based on the use of Fast Lyapunov Indicators modified by the introduction of a filtering window function, we compute sample three–dimensional representations of the manifolds which show an original vista about their complicated development in the phase-space.


      PubDate: 2016-03-11T12:45:40Z
       
  • Numerical simulation of surface waves instability on a homogeneous grid
    • Abstract: Publication date: Available online 10 March 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Alexander O. Korotkevich, Alexander I. Dyachenko, Vladimir E. Zakharov
      We performed full-scale numerical simulation of instability of weakly nonlinear waves on the surface of deep fluid. We show that the instability development leads to chaotization and formation of wave turbulence. Instability of both propagating and standing waves were studied. We separately studied pure capillary wave, that was unstable due to three-wave interactions and pure gravity waves, that were unstable due to four-wave interactions. The theoretical description of instabilities in all cases is included in the article. The numerical algorithm used in these and many other previous simulations performed by the authors is described in detail.


      PubDate: 2016-03-11T12:45:40Z
       
  • Border collisions inside the stability domain of a fixed point
    • Abstract: Publication date: Available online 24 February 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Viktor Avrutin, Zhanybai T. Zhusubaliyev, Erik Mosekilde
      Recent studies on a power electronic DC/AC converter (inverter) have demonstrated that such systems may undergo a transition from regular dynamics (associated with a globally attracting fixed point of a suitable stroboscopic map) to chaos through an irregular sequence of border-collision events. Chaotic dynamics of an inverter is not suitable for practical purposes. However, the parameter domain in which the stroboscopic map has a globally attracting fixed point has generally been considered to be uniform and suitable for practical use. In the present paper we show that this domain actually has a complicated interior structure formed by boundaries defined by persistence border collisions. We describe a simple approach that is based on symbolic dynamics and makes it possible to detect such boundaries numerically. Using this approach we describe several regions in the parameter space leading to qualitatively different output signals of the inverter although all associated with globally attracting fixed points of the corresponding stroboscopic map.


      PubDate: 2016-03-07T12:36:02Z
       
  • Hopf normal form with SN symmetry and reduction to systems of nonlinearly
           coupled phase oscillators
    • Abstract: Publication date: Available online 26 February 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Peter Ashwin, Ana Rodrigues
      Coupled oscillator models where N oscillators are identical and symmetrically coupled to all others with full permutation symmetry S N are found in a variety of applications. Much, but not all, work on phase descriptions of such systems consider the special case of pairwise coupling between oscillators. In this paper, we show this is restrictive - and we characterise generic multi-way interactions between oscillators that are typically present, except at the very lowest order near a Hopf bifurcation where the oscillations emerge. We examine a network of identical weakly coupled dynamical systems that are close to a supercritical Hopf bifurcation by considering two parameters, ϵ (the strength of coupling) and λ (an unfolding parameter for the Hopf bifurcation). For small enough λ > 0 there is an attractor that is the product of N stable limit cycles; this persists as a normally hyperbolic invariant torus for sufficiently small ϵ > 0 . Using equivariant normal form theory, we derive a generic normal form for a system of coupled phase oscillators with S N symmetry. For fixed N and taking the limit 0 < ϵ ≪ λ ≪ 1 , we show that the attracting dynamics of the system on the torus can be well approximated by a coupled phase oscillator system that, to lowest order, is the well-known Kuramoto-Sakaguchi system of coupled oscillators. The next order of approximation genericlly includes terms with up to four interacting phases, regardless of N . Using a normalization that maintains nontrivial interactions in the limit N → ∞ , we show that the additional terms can lead to new phenomena in terms of coexistence of two-cluster states with the same phase difference but different cluster size.


      PubDate: 2016-03-07T12:36:02Z
       
  • Traveling wave solutions in a chain of periodically forced coupled
           nonlinear oscillators
    • Abstract: Publication date: Available online 27 February 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): M. Duanmu, N. Whitaker, P.G. Kevrekidis, A. Vainchtein, J.E. Rubin
      Motivated by earlier studies of artificial perceptions of light called phosphenes, we analyze traveling wave solutions in a chain of periodically forced coupled nonlinear oscillators modeling this phenomenon. We examine the discrete model problem in its co-traveling frame and systematically obtain the corresponding traveling waves in one spatial dimension. Direct numerical simulations as well as linear stability analysis are employed to reveal the parameter regions where the traveling waves are stable, and these waves are, in turn, connected to the standing waves analyzed in earlier work. We also consider a two-dimensional extension of the model and demonstrate the robust evolution and stability of planar fronts. Our simulations also suggest the radial fronts tend to either annihilate or expand and flatten out, depending on the phase value inside and the parameter regime. Finally, we observe that solutions that initially feature two symmetric fronts with bulged centers evolve in qualitative agreement with experimental observations of phosphenes.


      PubDate: 2016-03-07T12:36:02Z
       
  • Filter accuracy for the Lorenz 96 model: Fixed versus adaptive observation
           operators
    • Abstract: Publication date: Available online 23 February 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): K.J.H. Law, D. Sanz-Alonso, A. Shukla, A.M. Stuart
      In the context of filtering chaotic dynamical systems it is well-known that partial observations, if sufficiently informative, can be used to control the inherent uncertainty due to chaos. The purpose of this paper is to investigate, both theoretically and numerically, conditions on the observations of chaotic systems under which they can be accurately filtered. In particular, we highlight the advantage of adaptive observation operators over fixed ones. The Lorenz ’96 model is used to exemplify our findings. We consider discrete-time and continuous-time observations in our theoretical developments. We prove that, for fixed observation operator, the 3DVAR filter can recover the system state within a neighbourhood determined by the size of the observational noise. It is required that a sufficiently large proportion of the state vector is observed, and an explicit form for such sufficient fixed observation operator is given. Numerical experiments, where the data is incorporated by use of the 3DVAR and extended Kalman filters, suggest that less informative fixed operators than given by our theory can still lead to accurate signal reconstruction. Adaptive observation operators are then studied numerically; we show that, for carefully chosen adaptive observation operators, the proportion of the state vector that needs to be observed is drastically smaller than with a fixed observation operator. Indeed, we show that the number of state coordinates that need to be observed may even be significantly smaller than the total number of positive Lyapunov exponents of the underlying system.


      PubDate: 2016-02-24T12:04:11Z
       
  • Editorial
    • Abstract: Publication date: 1 April 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 318–319
      Author(s): Falko Ziebert, Igor S. Aranson
      Nonlinear models are important to rationalize and understand self-organization, pattern formation and emergent behavior in molecular and cell biological systems. This special issue focuses on recent developments, that go beyond the classical modeling ideas of biochemical reactions and diffusion processes by including several effects identified recently as being crucial, for instance: elasticity/deformablity, anisotropy, multi-phase flow and ‘active’ behavior.


      PubDate: 2016-02-24T12:04:11Z
       
  • A modulation equations approach for numerically solving the moving soliton
           and radiation solutions of NLS
    • Abstract: Publication date: Available online 17 February 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Avy Soffer, Xiaofei Zhao
      Based on our previous work for solving the nonlinear Schrödinger equation with multichannel dynamics that is given by a localized standing wave and radiation, in this work we deal with the multichannel solution which consists of a moving soliton and radiation. We apply the modulation theory to give a system of ODEs coupled to the radiation term for describing the solution, which is valid for all times. The modulation equations are solved accurately by the proposed numerical method. The soliton and radiation are captured separately in the computation, and they are solved on the translated domain that is moving with them. Thus for a fixed finite physical domain in the lab frame, the multichannel solution can pass through the boundary naturally, which can not be done by imposing any existing boundary conditions. We comment on the differences of this method from the collective coordinates.


      PubDate: 2016-02-24T12:04:11Z
       
  • Stochastic shell models driven by a multiplicative fractional
           Brownian-motion
    • Abstract: Publication date: Available online 4 February 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Hakima Bessaih, María J. Garrido-Atienza, Björn Schmalfuss
      We prove existence and uniqueness of the solution of a stochastic shell–model. The equation is driven by an infinite dimensional fractional Brownian–motion with Hurst–parameter H ∈ ( 1 / 2 , 1 ) , and contains a non–trivial coefficient in front of the noise which satisfies special regularity conditions. The appearing stochastic integrals are defined in a fractional sense. First, we prove the existence and uniqueness of variational solutions to approximating equations driven by piecewise linear continuous noise, for which we are able to derive important uniform estimates in some functional spaces. Then, thanks to a compactness argument and these estimates, we prove that these variational solutions converge to a limit solution, which turns out to be the unique pathwise mild solution associated to the shell–model with fractional noise as driving process.


      PubDate: 2016-02-10T11:13:12Z
       
  • Forecasting turbulent modes with nonparametric diffusion models: Learning
           from noisy data
    • Abstract: Publication date: Available online 8 February 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Tyrus Berry, John Harlim
      In this paper, we apply a recently developed nonparametric modeling approach, the “diffusion forecast”, to predict the time-evolution of Fourier modes of turbulent dynamical systems. While the diffusion forecasting method assumes the availability of a noise-free training data set observing the full state space of the dynamics, in real applications we often have only partial observations which are corrupted by noise. To alleviate these practical issues, following the theory of embedology, the diffusion model is built using the delay-embedding coordinates of the data. We show that this delay embedding biases the geometry of the data in a way which extracts the most stable component of the dynamics and reduces the influence of independent additive observation noise. The resulting diffusion forecast model approximates the semigroup solutions of the generator of the underlying dynamics in the limit of large data and when the observation noise vanishes. As in any standard forecasting problem, the forecasting skill depends crucially on the accuracy of the initial conditions. We introduce a novel Bayesian method for filtering the discrete-time noisy observations which works with the diffusion forecast to determine the forecast initial densities. Numerically, we compare this nonparametric approach with standard stochastic parametric models on a wide-range of well-studied turbulent modes, including the Lorenz-96 model in weakly chaotic to fully turbulent regimes and the barotropic modes of a quasi-geostrophic model with baroclinic instabilities. We show that when the only available data is the low-dimensional set of noisy modes that are being modeled, the diffusion forecast is indeed competitive to the perfect model.


      PubDate: 2016-02-10T11:13:12Z
       
  • Autoresonance versus localization in weakly coupled oscillators
    • Abstract: Publication date: Available online 8 January 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Agnessa Kovaleva, Leonid I. Manevitch
      We study formation of autoresonance (AR) in a two-degree of freedom oscillator array including a nonlinear (Duffing) oscillator (the actuator) weakly coupled to a linear attachment. Two classes of systems are studied. In the first class of systems, a periodic force with constant (resonance) frequency is applied to a nonlinear oscillator (actuator) with slowly time-decreasing stiffness. In the systems of the second class a nonlinear time-invariant oscillator is subjected to an excitation with slowly increasing frequency. In both cases, the attached linear oscillator and linear coupling are time-invariant, and the system is initially engaged in resonance. This paper demonstrates that in the systems of the first type AR in the nonlinear actuator entails oscillations with growing amplitudes in the linear attachment while in the system of the second type energy transfer from the nonlinear actuator is insufficient to excite high-energy oscillations of the attachment. It is also shown that a slow change of stiffness may enhance the response of the actuator and make it sufficient to support oscillations with growing energy in the attachment even beyond the linear resonance. Explicit asymptotic approximations of the solutions are obtained. Close proximity of the derived approximations to exact (numerical) results is demonstrated.


      PubDate: 2016-01-12T00:45:22Z
       
  • Symmetry types and phase-shift synchrony in networks
    • Abstract: Publication date: Available online 31 December 2015
      Source:Physica D: Nonlinear Phenomena
      Author(s): Martin Golubitsky, Leopold Matamba Messi, Lucy E. Spardy
      In this paper we discuss what is known about the classification of symmetry groups and patterns of phase-shift synchrony for periodic solutions of coupled cell networks. Specifically, we compare the lists of spatial and spatiotemporal symmetries of periodic solutions of admissible vector fields to those of equivariant vector fields in the three cases of R n (coupled equations), T n (coupled oscillators), and ( R k ) n where k ≥ 2 (coupled systems). To do this we use the H / K Theorem of Buono and Golubitsky (2001) applied to coupled equations and coupled systems and prove the H / K theorem in the case of coupled oscillators. Josić and Török (2006) prove that the H / K lists for equivariant vector fields and admissible vector fields are the same for transitive coupled systems. We show that the corresponding theorem is false for coupled equations. We also prove that the pairs of subgroups H ⊃ K for coupled equations are contained in the pairs for coupled oscillators which are contained in the pairs for coupled systems. Finally, we prove that patterns of rigid phase-shift synchrony for coupled equations are contained in those of coupled oscillators and those of coupled systems.


      PubDate: 2016-01-04T00:20:31Z
       
  • Discrete synchronization of massively connected systems using hierarchical
           couplings
    • Abstract: Publication date: Available online 23 December 2015
      Source:Physica D: Nonlinear Phenomena
      Author(s): Camille Poignard
      We study the synchronization of massively connected dynamical systems for which the interactions come from the succession of couplings forming a global hierarchical coupling process. Motivations of this work come from the growing necessity of understanding properties of complex systems that often exhibit a hierarchical structure. Starting with a set of 2 n systems, the couplings we consider represent a two-by-two matching process that gather them in larger and larger groups of systems, providing to the whole set a structure in n stages, corresponding to n scales of hierarchy. This leads us naturally to the synchronization of a Cantor set of systems, indexed by { 0 , 1 } N , using the closed-open sets defined by n -tuples of 0 and 1 that permit us to make the link with the finite previous situation of 2 n systems: we obtain a global synchronization result generalizing this case. In the same context, we deal with this question when some defects appear in the hierarchy, that is to say when some couplings among certain systems do not happen at a given stage of the hierarchy. We prove we can accept an infinite number of broken links inside the hierarchy while keeping a local synchronization, under the condition that these defects are present at the N smallest scales of the hierarchy (for a fixed integer N ) and they be enough spaced out in those scales.


      PubDate: 2016-01-04T00:20:31Z
       
 
 
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